communicability¶

communicability(G)[source]¶

Return communicability between all pairs of nodes in G.

The communicability between pairs of nodes in G is the sum of closed walks of different lengths starting at node u and ending at node v.

Parameters:	G (graph) –
Returns:	comm – Dictionary of dictionaries keyed by nodes with communicability as the value.
Return type:	dictionary of dictionaries
Raises:	`NetworkXError` – If the graph is not undirected and simple.

See also

communicability_centrality_exp(): Communicability centrality for each node of G using matrix exponential.
communicability_centrality(): Communicability centrality for each node in G using spectral decomposition.
communicability(): Communicability between pairs of nodes in G.

Notes

This algorithm uses a spectral decomposition of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes $u$ and $v$ based on the graph spectrum is [1]

$C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},$

where $\phi_{j}(u)$ is the $u\rm{th}$ element of the $j\rm{th}$ orthonormal eigenvector of the adjacency matrix associated with the eigenvalue $\lambda_{j}$ .

References

[1]	Ernesto Estrada, Naomichi Hatano, “Communicability in complex networks”, Phys. Rev. E 77, 036111 (2008). http://arxiv.org/abs/0707.0756

Examples

>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability(G)